\[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
Test:
NMSE example 3.8
Bits:
128 bits
Bits error versus N
Time: 49.5 s
Input Error: 31.0
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\frac{\left(\left(1 + \frac{1}{N}\right) + \log N \cdot \log N\right) + \left(2 \cdot N\right) \cdot \left(\log N \cdot \log N\right)}{\log N \cdot N + \frac{1 + N}{1} \cdot \log \left(1 + N\right)} - \left(1 + \frac{\log N \cdot \left(\frac{\frac{2}{3}}{N} + \left(3 + 2 \cdot N\right)\right)}{\log N \cdot N + \frac{1 + N}{1} \cdot \log \left(1 + N\right)}\right)\)
  1. Started with
    \[\left(\left(N + 1\right) \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    31.0
  2. Using strategy rm
    31.0
  3. Applied flip-+ to get
    \[\left(\color{red}{\left(N + 1\right)} \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{\frac{{N}^2 - {1}^2}{N - 1}} \cdot \log \left(N + 1\right) - N \cdot \log N\right) - 1\]
    30.1
  4. Applied associate-*l/ to get
    \[\left(\color{red}{\frac{{N}^2 - {1}^2}{N - 1} \cdot \log \left(N + 1\right)} - N \cdot \log N\right) - 1 \leadsto \left(\color{blue}{\frac{\left({N}^2 - {1}^2\right) \cdot \log \left(N + 1\right)}{N - 1}} - N \cdot \log N\right) - 1\]
    29.9
  5. Applied simplify to get
    \[\left(\frac{\color{red}{\left({N}^2 - {1}^2\right) \cdot \log \left(N + 1\right)}}{N - 1} - N \cdot \log N\right) - 1 \leadsto \left(\frac{\color{blue}{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}}{N - 1} - N \cdot \log N\right) - 1\]
    29.9
  6. Using strategy rm
    29.9
  7. Applied flip-- to get
    \[\color{red}{\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1} - N \cdot \log N\right)} - 1 \leadsto \color{blue}{\frac{{\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1}\right)}^2 - {\left(N \cdot \log N\right)}^2}{\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1} + N \cdot \log N}} - 1\]
    29.9
  8. Applied simplify to get
    \[\frac{{\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1}\right)}^2 - {\left(N \cdot \log N\right)}^2}{\color{red}{\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1} + N \cdot \log N}} - 1 \leadsto \frac{{\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1}\right)}^2 - {\left(N \cdot \log N\right)}^2}{\color{blue}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)}} - 1\]
    29.9
  9. Applied taylor to get
    \[\frac{{\left(\frac{\left({N}^2 - 1\right) \cdot \log \left(N + 1\right)}{N - 1}\right)}^2 - {\left(N \cdot \log N\right)}^2}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)} - 1 \leadsto \frac{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)} - 1\]
    0.0
  10. Taylor expanded around inf to get
    \[\frac{\color{red}{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)} - 1 \leadsto \frac{\color{blue}{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)} - 1\]
    0.0
  11. Applied simplify to get
    \[\frac{\left(1 + \left({\left(\log N\right)}^2 + \left(\frac{1}{N} + 2 \cdot \left({\left(\log N\right)}^2 \cdot N\right)\right)\right)\right) - \left(2 \cdot \left(\log N \cdot N\right) + \left(3 \cdot \log N + \frac{2}{3} \cdot \frac{\log N}{N}\right)\right)}{\log N \cdot N + \frac{N + 1}{1} \cdot \log \left(N + 1\right)} - 1 \leadsto \frac{1 + \left(\left(\left(2 \cdot N\right) \cdot \log N\right) \cdot \log N + \left(\frac{1}{N} + \log N \cdot \log N\right)\right)}{N \cdot \log N + \log \left(1 + N\right) \cdot \frac{1 + N}{1}} - \left(\frac{\frac{\log N}{\frac{N}{\frac{2}{3}}} + \left(\left(2 \cdot N\right) \cdot \log N + \log N \cdot 3\right)}{N \cdot \log N + \log \left(1 + N\right) \cdot \frac{1 + N}{1}} + 1\right)\]
    0.4
  12. Applied final simplification
  13. Applied simplify to get
    \[\color{red}{\frac{1 + \left(\left(\left(2 \cdot N\right) \cdot \log N\right) \cdot \log N + \left(\frac{1}{N} + \log N \cdot \log N\right)\right)}{N \cdot \log N + \log \left(1 + N\right) \cdot \frac{1 + N}{1}} - \left(\frac{\frac{\log N}{\frac{N}{\frac{2}{3}}} + \left(\left(2 \cdot N\right) \cdot \log N + \log N \cdot 3\right)}{N \cdot \log N + \log \left(1 + N\right) \cdot \frac{1 + N}{1}} + 1\right)} \leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{N}\right) + \log N \cdot \log N\right) + \left(2 \cdot N\right) \cdot \left(\log N \cdot \log N\right)}{\log N \cdot N + \frac{1 + N}{1} \cdot \log \left(1 + N\right)} - \left(1 + \frac{\log N \cdot \left(\frac{\frac{2}{3}}{N} + \left(3 + 2 \cdot N\right)\right)}{\log N \cdot N + \frac{1 + N}{1} \cdot \log \left(1 + N\right)}\right)}\]
    0.4
  14. Removed slow pow expressions

Original test:


(lambda ((N default))
  #:name "NMSE example 3.8"
  (- (- (* (+ N 1) (log (+ N 1))) (* N (log N))) 1)
  #:target
  (- (log (+ N 1)) (- (/ 1 (* 2 N)) (- (/ 1 (* 3 (sqr N))) (/ 4 (pow N 3))))))